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Matrices whose coefficients are linear forms in logarithms. (English) Zbl 0763.11030
Denote by $$L$$ the $${\mathbb{Q}}$$-vector space of complex numbers $$\ell$$ such that $$e^{\ell}$$ is an algebraic number, and by $${\mathcal L}$$ the vector space generated by $$1$$ and $$L$$ over the field $$\overline\mathbb{Q}$$ of algebraic numbers. The elements of $$L$$ are the logarithms of non-zero algebraic numbers, while the elements of $${\mathcal L}$$ are the linear forms in logarithms of algebraic numbers: $$\beta_{0}+\beta_{1}\log\alpha_{1}+\cdots+ \beta_{n}\log\alpha_{n}$$. One result of this paper is a lower bound for the rank of a matrix whose entries are in $${\mathcal L}$$.
The simplest case is as follows: a $$2\times3$$ matrix with entries in $${\mathcal L}$$ whose columns are $$\overline\mathbb{Q}$$-linearly independent, and whose rows are $${\overline\mathbb{Q}}$$-linearly independent, has rank 2 (the six exponentials theorem deals with a $$2\times3$$ matrix with entries in $$L$$). The proof relies on the theorem of the linear subgroup, which is sharpened by the author; this refinement is performed by means of arguments where the author uses the language of category theory. He deduces improvements of earlier results due to M. Emsalem [C. R. Acad. Sci., Paris, Sér. I 297, 225-227 (1983; Zbl 0529.12006); J. Reine Angew. Math. 382, 181-198 (1987; Zbl 0621.12008)] and M. Laurent [J. Reine Angew. Math. 399, 81-108 (1989; Zbl 0666.12001)] on the conjectures of Leopoldt and Jaulent concerning the $$p$$-adic rank of $$S$$- units in a number field.

##### MSC:
 11J81 Transcendence (general theory) 11R27 Units and factorization
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##### References:
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